Mathematics for the interested outsider

Intertwinors from Semistandard Tableaux Span, part 1

Now that we’ve shown the intertwinors that come from semistandard tableaux are independent, we want to show that they span the space . This is a bit fidgety, but should somewhat resemble the way we showed that standard polytabloids span Specht modules.

First of all, I say that if and , then the coefficients of and differ by a factor of . Indeed, we calculate

This tells us that

Comparing coefficients on the left and right gives us our assertion.

As an immediate corollary to this lemma, we conclude that if has a repetition in some column, then . Indeed, we can let be the permutation that swaps the places of these two identical entries. Then , while the previous result tells us that , and so .

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